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On the theorem of Belyi. (À propos du théorème de Belyi.) (French) Zbl 0869.11092

A famous theorem of Belyi asserts that on any smooth projective geometrically connected algebraic curve \(C\) defined over \(\overline\mathbb{Q}\) there exists a function \(f:C\to \mathbb{P}^1\) unramified outside \(\{0,1, \infty\}\). The author shows that this function \(f\) can be chosen without nontrivial automorphism, i.e. if \(\sigma\) is an automorphism of \(C\) with \(f\circ \sigma=f\) then \(\sigma=1\). As a consequence, for \(K\subset \mathbb{C}\) a finite extension of \(\mathbb{Q}\), anany \(K\)-isomorphism class of smooth projective geometrically connected algebraic curves can be characterized by a “dessin d’enfant de Grothendieck”, i.e. an isomorphism class of finite connected topological coverings of the sphere minus three points.
Reviewer: N.Vila (Barcelona)

MSC:

11R32 Galois theory
11G30 Curves of arbitrary genus or genus \(\ne 1\) over global fields
14H30 Coverings of curves, fundamental group
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References:

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[3] Jean-Marc, Couveignes, Calcul et rationalité de fonctions de Belyi en genre 0, Annales de l’Institut Fourier44 (1994), 1-38. · Zbl 0791.11059
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